Computing Qualitatively Correct Approximations of Balance Laws

The book gives an overview of recent numerical techniques for the integration of partial differential equations, especially hyperbolic systems of balance laws in one space dimension (Part I) and weakly nonlinear kinetic equations (Part II). Several of its salient features are:

• Surveys both analytical and numerical aspects of hyperbolic balance laws (including the recent theory of viscosity solutions for systems)
• Numerous derivations of both well-balanced and asymptotic-preserving schemes emphasizing relations between each other Includes original material about K-multibranch solutions for linear geometric optics or order-preserving strings
• Several chapters about numerical approximation of chemotaxis or semiconductor kinetic models which display constant macroscopic fluxes at stationary state ("qualitatively correct" approximations)
• Presents well-balanced techniques for linearized Boltzmann and Fokker-Planck kinetic equations relying on "Caseology" methods.

Laurent Gosse received the M.S. and Ph.D. degrees both in Mathematics from Universities of Lille 1 and Paris IX Dauphine in 1991 and 1997 respectively. Between 1997 and 1999 he was a TMR postdoc in IACM-FORTH (Heraklion, Crete) mostly working on well-balanced numerical schemes and a posteriori error estimates with Ch. Makridakis. From 1999 to 2001, he was postdoc in Universtity of L'Aquila (Italy) working on stability theory for systems of balance laws and multiphase computations in geometrical optics with K-multibranch solutions. In 2001, he moved to University of Pavia working on asymptotic-preserving schemes and degenerate parabolic equations with G. Toscani. In 2002, he was granted a permanent researcher position at CNR in Bari (Italy) where he developed Lagrangian schemes for nonlinear diffusion models and a stable inversion algorithm for Markov moment problem with O. Runborg. Numerical investigation of semiclassical WKB approximation for quantum models of crystals was conducted with P.A. Markowich between 2003 and 2006. Since 2011, he holds a CNR position at both Roma and University of L'Aquila and works mainly on the applications of Caseology to well-balanced schemes for collisional kinetic equations.